Low-speed correction method for Stribeck friction model of mechanical arm

Abstract

A mechanical arm Stribeck friction model low-speed correction method comprises the steps that a friction response function subjected to speed normalization correction is introduced, a Stribeck model subjected to low-speed correction is established, and the Stribeck model is applied to a GMS friction model; a step-by-step parameter identification method is adopted and comprises the following steps of: firstly, identifying a static friction parameter phi RSFStribeck of a low-speed correction Stribeck model on the basis of multiple groups of constant-speed moving tracks; and then, embedding the correction model into a generalized Maxwell slip GMS model framework, freezing the identified static parameters, selecting a triangular wave displacement signal as an excitation track, and further identifying the dynamic friction parameter phi GMS of the GMS model. According to the method, external interference of a static model can be inhibited, the identification precision of the model is improved, meanwhile, the precision is further improved in a dynamic model, and high-precision continuous characterization of the friction model is achieved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of mechanical arm friction identification, and aims at the problem that the friction model is not accurate enough in low-speed high-precision actual occasions. BACKGROUND

[0002] With the intelligentization and modernization of China's robot industry, the requirements for robot motion control performance continue to improve. In low-speed high-precision application scenarios such as automobile manufacturing, medical surgery and precision machining, the trajectory tracking accuracy and stability of the mechanical arm are directly affected by the joint friction. In order to realize high-precision force control of a six-degree-of-freedom mechanical arm, improve human-machine interaction efficiency and meet high-end industrial needs, accurate identification of joint friction becomes a key problem. Therefore, scholars have proposed various static models and dynamic models to describe the friction phenomenon. Static models such as Coulomb friction model, Coulomb + viscous friction model and Stribeck friction model, etc. Among them, Stribeck friction can describe the characteristics that friction force decreases first and then rises with the increase of speed, but it still cannot comprehensively cover the friction phenomenon. Then, scholars proposed dynamic models such as Lugre model, GMS (Generalized Maxwell-Slip) friction model, etc. based on the Stribeck friction model, which introduced internal state variables through differential equations to represent the pre-sliding friction hysteresis and dynamic transition behavior, but the parameter identification complexity and low-speed fitting accuracy still have limitations. SUMMARY

[0003] In order to overcome the shortcomings of the existing friction model in low-speed prediction inaccuracy caused by the description deviation of Stribeck effect, the application proposes a low-speed modified Stribeck friction model and its application method in GMS model: through the low-speed convergence mechanism in the Stribeck curve, a speed normalized modified friction response function is introduced, and embedded into the pre-sliding hysteresis framework of the GMS friction model, realizing high-precision continuous representation of friction, and significantly improving the stability of low-speed motion control.

[0004] The application provides the following solutions to solve the above technical problems:

[0005] A low-speed modified Stribeck friction model for a mechanical arm, comprising the following steps:

[0006] Step 1, design joint motion of friction sliding interval;

[0007] Step 2, low-speed modified Stribeck model parameter identification, process is:

[0008] 2.1 Low-speed modified Stribeck model modeling;

[0009] 2.2 Parameter identification: for the parameter identification problem of the low-speed modified Stribeck model, a genetic algorithm is used as the core identification method to identify the parameters Φ RSF_Stribeck c S S m R1, R2}, and particle swarm optimization PSO, dream optimization algorithm DOA and marine predator algorithm MPA are introduced for comparison and verification. By setting the root mean square of residual as the objective function, the performance of each algorithm is quantitatively evaluated to verify the effectiveness and robustness of the model.

[0010] Step 3, parameter identification of the low-speed modified Stribeck model combined with the GMS friction model, the process is:

[0011] 3.1 Acquisition of pre-sliding friction data;

[0012] 3.2 GMS friction model combined with low-speed modified Stribeck modeling:

[0013] 3.3 Parameter identification of the GMS friction model: first, identify the parameters of the low-speed modified Stribeck model, then lock the identified parameters, and identify the parameters Φ GMS i i of the GMS friction model. The genetic algorithm is used for parameter estimation, the weighted root mean square WRMSE is used as the objective function, and the low-speed region and the speed reversal region are the core concerns.

[0014] Further, the process of step 1 is:

[0015] 1.1 Lock other non-detection joints, move the 6th joint to the vertical part, so that the symmetric axis of the joint is parallel to the direction of the robot arm gravity;

[0016] 1.2 In order to make the friction identification more sufficient, this method designs uniform velocity trajectories in different speed intervals, and increases the weight of joint movement in the low-speed region. The joint speed range is determined to be 0.05(°) / s to 35(°) / s, and a total of 51 uniform velocity motion trajectories in the positive and negative directions of the joint are designed.

[0017] 1.3 Process the speed-torque data. Since the axis of joint i is consistent with the direction of gravity, the input torque of joint i only includes friction torque, and the inertia force determined by angular acceleration is zero.

[0018] Preferably, in 1.3, if the joint i is not parallel, the following steps are performed:

[0019] ​​​​​​1) Drive joint i to move at constant velocity from θ0 to θ1 and then from θ1 back to θ0 at constant velocity;

[0020] 2) During forward motion, joint input torque τ1 is

[0021]

[0022] where G(θ) is the gravity torque, is the friction torque;

[0023] During reverse motion, joint input torque τ2 is:

[0024]

[0025] The robot arm uses a harmonic reducer with high motion precision, and the forward and reverse friction forces are approximately equal, resulting in the following equation:

[0026]

[0027] From (1) to (3), the following formula for the friction term and the gravity term is obtained:

[0028]

[0029] Therefore, the friction torque is extracted from the uniform forward and reverse motion of the joint, and a set of velocity-friction sequences is obtained.

[0030] Further, in 2.1, the 2.1 low-speed modified Stribeck model is modeled, which expands the classic Stribeck friction model into the following form:

[0031]

[0032] RSF(t) is the friction response shape function, and the formula is as follows:

[0033]

[0034] where, F c is the Coulomb friction; F s is the static friction; v s is the Stribeck speed; δ is the exponential coefficient; R1, R2 are response state parameters; σ is the viscosity coefficient; t is the speed normalization parameter, specifically:

[0035]

[0036] v m is the shape characteristic speed.

[0037] The introduction of a state correction coefficient (1+RSF(t)) aims to suppress the zero-drift phenomenon of the robotic arm. When the robotic arm starts from zero speed, the frictional force in the low-speed range often shifts due to external disturbances such as temperature changes and surface lubrication conditions. This improved model can effectively suppress such shifting phenomena.

[0038] At the microscopic scale, when |v| < v m At low speeds, RSF(t) modulates the frictional behavior. When v approaches 0, the zero-velocity response of the frictional force is dominated by R1, altering the initial attenuation characteristics of the frictional force. Conversely, when v approaches v... m When this occurs, the frictional force is primarily influenced by R2. Typically, v m To characterize the characteristic velocities near the Stribeck effect, corresponding to the velocity range where friction rises to its maximum value in the low-speed region. This region exhibits a large gradient in frictional force change and significant noise, making it difficult to identify accurately. This model introduces R² to adjust the local shape of the model for more precise identification.

[0039] On a macroscopic scale, when |v|≥v m When t takes the value 1, the model degenerates into the original Stribeck model.

[0040] Furthermore, in section 2.2, the objective function is as follows:

[0041]

[0042] Among them, F f,i The friction force data is obtained from the i-th sample. It is the predicted friction force after the i-th model is identified, and K is the total number of friction force samples.

[0043] In section 3.1,

[0044] A triangular wave displacement signal is selected as the input trajectory to ensure that the motion is in a pre-sliding state. The formula for the triangular wave displacement signal is as follows:

[0045]

[0046] Where A is the maximum angle of the trajectory and T is the motion period;

[0047] The collected friction data were filtered using the Robust Locally Weighted Scatter Smoothing (RLOESS) method to better preserve the nonlinear characteristics of friction and reduce time shift.

[0048] In section 3.2, considering the case of viscous friction, the GMS model expression is as follows:

[0049]

[0050] Wherein, N is the number of equivalent parallel friction units; F i is the friction force of a single friction unit; σ is the viscous friction coefficient; and v is the velocity.

[0051] 3.2.1) In the pre-sliding state, the friction force exhibits a displacement-dependent mechanical behavior, and the state equation is

[0052]

[0053] Wherein, k i is the stiffness coefficient of the friction unit, representing the elastic response characteristics in the pre-sliding stage.

[0054] 3.2.2) When α i · f (v) > F i , the system exits the viscous state and enters the sliding friction state, and the state equation is as follows:

[0055] Wherein, C is the attraction parameter, determining the convergence rate of the friction force in the sliding stage.

[0056] In order to accurately describe the friction characteristics in the low-speed region, a modified Stribeck friction function s f is introduced as follows:

[0057]

[0058] Wherein, the definitions of the parameter term β(t) and the Stribeck model parameters and their variation rules are given in the aforementioned formulas (6) to (8).

[0059] In the aforementioned 3.3, the objective function and the weighting mechanism are as follows:

[0060]

[0061] Wherein, w i is the weighting weight, v th represents the low-speed velocity threshold, and Δsign(v i ) represents the speed reversal event.

[0062] The technical concept of the present application is: in view of the problem that the existing friction model causes prediction accuracy to decrease due to the description deviation of Stribeck effect caused by temperature, lubrication and other interference factors in the low-speed working condition, a Stribeck modeling method based on speed normalization correction is proposed.

[0063] The application has the beneficial effects that the identification accuracy of the friction model is significantly improved, the structure of the static friction model is optimized, and the static friction model is successfully embedded in the dynamic model framework, the identification accuracy of the friction model is improved under the dynamic GMS friction model, the model adaptability in the Stribeck low-speed area is enhanced by introducing the speed normalization correction friction response function, and the stability of the system against external disturbance is effectively improved, the Stribeck effect and the friction hysteresis characteristics are accurately characterized by the model through the friction response function, and excellent dynamic response characteristics are exhibited in the continuous transition of the pre-sliding and sliding states. BRIEF DESCRIPTION OF DRAWINGS

[0064] Figure 1 The identification effect diagram of the classic Stribeck model.

[0065] Figure 2 The identification effect diagram of the low-speed correction Stribeck model.

[0066] Figure 3 The identification effect diagram of the classic GMS friction model.

[0067] Figure 4 The identification effect diagram of the classic GMS friction model.

[0068] Figure 5 The identification effect diagram of the low-speed correction Stribeck embedded GMS friction model in the displacement-friction force scale.

[0069] Figure 6 The identification effect diagram of the low-speed correction Stribeck embedded GMS friction model in the time-friction force scale.

[0070] Figure 7 The flowchart of the application. DETAILED DESCRIPTION

[0071] The application will be further described below with reference to the drawings.

[0072] Reference Figures 1-6 A low-speed correction method of a Stribeck friction model of a mechanical arm, comprising the following steps:

[0073] Step 1, joint movement of the friction sliding interval is designed, and the process is as follows:

[0074] 1.1 lock other non-detection joints, move the 6th joint to the vertical part, make the symmetric central axis of the joint parallel to the direction of the gravity of the mechanical arm;

[0075] 1.2 In order to make the friction identification more sufficient, the method designs uniform speed trajectories in different speed intervals, and increases the weight of low speed interval joint motion, the joint speed range is 0.05(°) / s to 35(°) / s, the specific scheme is 0.05; 0.1; 0.15; 0.2; 0.25; 0.3; 0.35; 0.4; 0.45; 0.5; 0.55; 0.6; 0.65; 0.7; 0.75; 0.8; 0.85; 0.9; 0.95; 1.0; 1.1; 1.2; 1.3; 1.4; 1.5; 1.6; 1.7; 1.8; 1.9; 2; 2.2; 2.4; 2.6; 2.8; 3.0; 3.4; 3.8; 4.2; 4.6; 5.0; 6; 7; 8; 9; 10; 14; 18; 22; 26; 30; 35(the above speed unit is(°) / s)Each group of speed needs to be moved in the forward and reverse directions, and the trajectory period is 10s;

[0076] 1.3 Process the speed-torque data: since the axis of joint i is consistent with the direction of gravity, the input torque of joint i only includes friction torque, and the inertia force determined by angular acceleration is zero, if joint i is not in parallel condition, the following steps need to be performed:

[0077] 1.3.1) drive joint i to move at a uniform speed from θ0 to θ1, and then return to θ0 at a uniform speed from θ1;

[0078] 1.3.2) during the forward motion, the joint input torque τ1 is

[0079]

[0080] wherein G(θ) is the gravity torque, is the friction torque;

[0081] and during the reverse motion of the speed, the joint input torque τ2 is:

[0082]

[0083] The mechanical arm adopts a harmonic reducer with high motion precision, and the forward and reverse friction forces are approximately equal, and the following formula is obtained:

[0084]

[0085] From (1) to (3), the following formula of the friction term and the gravity term is obtained:

[0086]

[0087] Therefore, the friction torque is extracted from the uniform speed forward and reverse motion of the joint, and a set of speed-friction sequences is obtained.

[0088] Step 2, low-speed correction Stribeck model parameter identification, the process is:

[0089] 2.1 Low-speed correction Stribeck model modeling, the classical Stribeck friction model is extended to the following form:

[0090]

[0091] In the formula, RSF(t) is the friction response shape function, and the formula is as follows:

[0092]

[0093] Where, F c is the Coulomb friction; F s is the static friction; v s is the Stribeck speed; δ is the exponential coefficient; R1, R2 are response state parameters; σ is the viscosity coefficient; t is the speed normalized parameter, specifically:

[0094]

[0095] v m is the shape characteristic speed;

[0096] The state correction coefficient (1+RSF(t)) is introduced to suppress the zero drift phenomenon of the robot arm. When the robot arm starts from zero speed, the friction force in the low speed section will often deviate due to the influence of external disturbances such as temperature change and surface lubrication condition. The improved model can effectively suppress such deviation phenomenon.

[0097] In the microscale, when |v|<v m , RSF(t) will modulate the low-speed friction shape. When v tends to 0, the zero-speed response of the friction force is dominated by R1, and the initial decay characteristics of the friction force are changed. When v tends to v m , the friction force is dominated by R2. Usually v m is a characteristic speed near which the Stribeck effect occurs, corresponding to the speed interval near which the low-speed friction rises to the maximum value. The friction force changes greatly in this region, and the noise is significant, which is not easy to accurately identify. The model introduces R2 to adjust the local shape of the model to accurately identify.

[0098] In the macro scale, when |v|≥v m , t takes the value of 1. The model becomes degenerate into the original Stribeck model.

[0099] 2.2 Parameter identification: for the parameter identification problem of the low-speed corrected Stribeck model, the genetic algorithm (Genetic Algorithm, GA) is used as the core identification method, and the parameters ΦRSF_Stribeck = {F c , F S , v S , δ, σ, v m , R1, R2} are identified, and particle swarm optimization (PSO), dream optimization algorithm (DOA) and marine predator algorithm (MPA) are introduced for comparison and verification. The root mean square of residual error is set as the objective function to quantitatively evaluate the performance of each algorithm, and the population size of each algorithm is set to 100 and the maximum number of iterations is set to 3000 to verify the effectiveness and robustness of the model.

[0100] The objective function is as follows:

[0101]

[0102] where F f,i is the friction data obtained by the i-th sampling, is the predicted friction after identification of the i-th model, and K is the total amount of friction samples.

[0103] The finally identified parameters F c = 1.1068 N·m; F s = 3.5378 N·m; v s = 13.4605 (°) / s; δ = 0.6091; σ = 0.1126 N·m; v m = 0.6663 (°) / s; R1 = -0.5380; R2 = 1.0716;

[0104] Step 3, low-speed modified Stribeck model combined with GMS friction model parameter identification, process is:

[0105] 3.1 Acquisition of pre-sliding friction data

[0106] Select a triangular wave displacement signal as the input trajectory to ensure that the movement is in the pre-sliding state. The formula of the triangular wave displacement signal is as follows:

[0107]

[0108] where A is the maximum angle of the trajectory, and T is the movement period.

[0109] The collected friction data is filtered by robust local weighted scatter smoothing method (RLOESS) with a smoothing coefficient of 0.06 to better preserve the nonlinear characteristics of friction and reduce time shift;

[0110] 3.2 GMS friction model combined with low-speed modified Stribeck modeling:

[0111] Considering the case of viscous friction, the expression of GMS model is as follows:

[0112]

[0113] where N is the number of equivalent parallel friction units; F i is the friction force of a single friction unit; σ is the viscous friction coefficient; v is the velocity;

[0114] 3.2.1) In the pre-sliding state, the friction force exhibits a displacement-dependent mechanical behavior, and the state equation is

[0115]

[0116] where k i is the stiffness coefficient of the friction unit, representing the elastic response characteristics in the pre-sliding stage;

[0117] 3.3.2) When α i · f (v) > F i , the system exits the viscous state and enters the sliding friction state, and the state equation is as follows:

[0118]

[0119] where C is the attraction parameter, determining the convergence rate of the friction force in the sliding stage;

[0120] To accurately describe the friction characteristics in the low-speed region, a modified Stribeck friction function s f is introduced as follows:

[0121]

[0122] where the parameter term β(t) and the definition of the Stribeck model parameters and their variation are given in the aforementioned equations (6) to (8).

[0123] 3.3 GMS friction model parameter identification: When identifying the relevant parameters of the GMS friction model, first, the low-speed modified Stribeck model parameters are identified, and then the identified parameters Φ RSF_Stribeck are locked. Subsequently, the GMS friction model parameters Φ GMS = {C, k i , α i} are identified; the identification method still uses the genetic algorithm for parameter estimation. In order to make the identification effect more accurate, the weighted residual mean square WRMSE is used to design the objective function, and the core focuses on the low-speed region and the speed reversal region. The objective function and the weighting mechanism are as follows:

[0124]

[0125] where wi is the weighted weight, v th represents the low-speed velocity threshold, Δsign(v i ) represents the velocity reversal event;

[0126] The method selects N = 3 parallel friction units; the final identification results are k1 = 3599.6989 N·m / (°), k2 = 1012.6772 N·m / (°), and k3 = 225.1308 N·m / (°); α1 = 0.1905, α2 = 0.1840, and α3 = 0.6265; and C = 2.4791 N·m / (°).

[0127] To verify the effectiveness of the method, Figure 1 and Figure 2 respectively show the parameter identification results of the sliding friction interval. Figure 1 The identification effect of the classical Stribeck model is shown, and the RMSE value obtained by the genetic algorithm is 0.2038 N·m; Figure 2 is the identification effect of the low-speed modified Stribeck model, and the RMSE value obtained by the genetic algorithm is 0.0820 N·m. The final data show that in the zero-speed vicinity area, the low-speed modified Stribeck model significantly improves the fitting degree of the friction force prediction curve and the actual friction force data, and its output has higher smoothness and continuity. In particular, the model exhibits more accurate fitting characteristics in the maximum static friction force area.

[0128] To further demonstrate the effectiveness of the low-speed modified Stribeck friction model, the residual mean square error RMSE values of the identification of each identification method are given below, and Table 1 shows the identification values of different optimization algorithms:

[0129] Algorithm GA PSO DOA MPA RMSE 0.0820 N-m 0.0823 N-m 0.0824 N-m 0.0820 N-m

[0130] Table 1

[0131] Through system verification by various optimization algorithms, the results show that the model maintains stable performance under different identification methods, proving the universality of the model.

[0132] Figures 3 to 6 Further comparison of the dynamic characteristic representation ability of the classical GMS friction model and the GMS friction model embedded with the low-speed modified Stribeck module. In the displacement-friction force phase plane ( Figure 3 , Figure 4 ) and time-friction force time domain response ( Figure 5 , Figure 6On the scale, the low-speed modified Stribeck embedded GMS friction model presents superior identification performance. Among them, the classic GMS friction model is 0.2110N·m on the WRMSE index, and the GMS friction model embedded with low-speed modified Stribeck is improved to 0.1927N·m, which quantitatively verifies the continuous optimization ability of the strategy in the dynamic friction model.

[0133] The above describes the excellent optimization effect of an embodiment given by the application. Obviously, the application is not limited to the above embodiment, and various modifications can be made to the application without departing from the basic spirit of the application and without exceeding the scope involved by the essential content of the application.

Claims

1. A method for low speed correction of a Stribeck friction model of a robot arm, characterized in that, The method comprises the following steps: Step 1, design the joint motion of the friction sliding interval; Step 2, low-speed modified Stribeck model parameter identification, the process is: 2.1 low-speed modified Stribeck model modeling; 2.2 Parameter identification: For the parameter identification problem of the low-speed modified Stribeck model, genetic algorithm is used as the core identification method to identify the parameters Φ RSF_Stribeck ={F c ,F S ,ν S ,δ,σ,v m ,R1,R2}, and particle swarm optimization (PSO), dream optimization algorithm (DOA), and marine predator algorithm (MPA) are introduced for comparison and verification. By setting the root mean square of residual as the objective function, the performance of each algorithm is quantitatively evaluated to verify the effectiveness and robustness of the model; Step 3, low-speed modified Stribeck model combined with GMS friction model parameter identification, the process is: 3.1 acquisition of pre-sliding friction data; 3.2 GMS friction model combined with low-speed modified Stribeck modeling: 3.3 GMS friction model parameter identification: first, identify the low-speed modified Stribeck parameters, then lock the identified parameters, and identify the GMS friction model parameters Φ GMS = {C, k i , α i} . The identification method uses genetic algorithm for parameter estimation, the objective function uses weighted residual mean square WRMSE design, and the core focuses on the low-speed area and the speed reversal area.

2. The method of claim 1, wherein the Stribeck friction model is modified at low speeds by, The process of step 1 is: 1.1 lock other non-detection joints, move the 6th joint to the vertical part, make the symmetric axis of the joint parallel to the direction of the mechanical arm gravity; 1.2 design uniform velocity trajectories in different speed intervals, and increase the weight of low-speed interval joint motion to determine the joint speed range and design the uniform velocity motion trajectory in the positive and negative directions of the joint; 1.3 process the speed-torque data: since the axis of joint i is consistent with the direction of gravity, the input torque of joint i only includes friction torque, and the inertia force determined by angular acceleration is zero.

3. The method of claim 2, wherein the Stribeck friction model is modified at low speeds by, In 1.3, if joint i is not in parallel condition, the following steps are performed: 1.3.1) drive joint i to move at a uniform speed from θ0 to θ1, and then return to θ0 at a uniform speed from θ1; 1.3.2) during the positive motion, the joint input torque τ1 is where G(0) is the gravity moment, is the friction moment; And during the reverse motion, the joint input torque τ2 is: The mechanical arm adopts a harmonic reducer with high motion accuracy, and the positive and negative friction forces are approximately equal, so the following formula is obtained: From (1) to (3), the following formula of friction term and gravity term is obtained: The friction torque is extracted from the uniform speed positive and negative rotation of the joint to obtain a set of speed-friction sequences.

4. The method of claim 3, wherein the Stribeck friction model is modified at low speeds by, In 2.1, the classical Stribeck friction model is expanded into the following form: RSF(t) is the friction response shape function, and the formula is as follows: RSF(t) = (1 - t) 3 R1 + t(1 - t) 2 R2 (7) where F c is Coulomb friction; F s is static friction; v s is Stribeck velocity; δ is an exponent coefficient; R1, R2 are response state parameters; σ is a viscosity coefficient; t is a velocity normalized parameter, in particular: where v m is the morphological characteristic velocity.

5. The method of claim 4, wherein the Stribeck friction model is modified at low speeds by, In 2.2, the objective function is as follows: where F f,i is the friction force data obtained by the i-th sampling, F fpred,i is the predicted friction force after the i-th model identification, and K is the total amount of friction force samples.

6. The method of claim 5, wherein the Stribeck friction model is modified at low speeds by, In 3.1, a triangular wave displacement signal is selected as the input trajectory to ensure that the motion is in the pre-sliding state, and the formula of the triangular wave displacement signal is as follows: Where A is the maximum angle of the trajectory, and T is the motion period; The collected friction data is filtered by using the robust local weighted scatter smoothing method RLOESS to better retain the nonlinear characteristics of the friction and reduce the time shift.

7. The method of claim 6, wherein the Stribeck friction model is modified at low speeds by, In 3.2, considering the case of viscous friction, the expression of GMS model is as follows: where N is the number of equivalent parallel friction units; F i is the friction force of a single friction unit; σ is the viscous friction coefficient; and v is the velocity. 3.2.1) in the pre-sliding state, the friction force shows a displacement-related mechanical behavior, and the state equation is In the formula, k i is the friction unit stiffness coefficient, representing the elastic response characteristics of the pre-sliding stage; 3.2.2) When a i f (v) > F i The system leaves the stick state and enters the sliding friction state, and the state equation is as follows:​ where C is an attraction parameter that determines the convergence rate of the friction force in the sliding phase; a i is a weight coefficient; To precisely describe the friction characteristics in the low speed region, a modified Stribeck friction function s f is introduced as follows:

8. The method of claim 7, wherein the Stribeck friction model is modified at low speeds by, In 3.3, the objective function and the weighting mechanism are as follows respectively: where ω i is a weighting weight, v th represents a low speed velocity threshold, Δsign(v i ) represents a velocity reversal event.

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